A079253 -id:A079253 - cp's OEIS frontend

A080720 a(0) = 5; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 3".

5, 7, 8, 10, 11, 12, 13, 15, 18, 19, 21, 24, 27, 30, 31, 33, 34, 35, 36, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 54, 55, 57, 60, 63, 66, 67, 68, 69, 72, 73, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 103, 104, 105, 108, 109, 111, 112, 113, 114, 115, 116, 117, 118, 119

Offset: 0

N. J. A. Sloane, Mar 05 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)
Index entries for sequences of the a(a(n)) = 2n family

Cf. A079000, A003605, A079253, A080711, A080710.

{a=5; m=[5]; for(n=1,66,print1(a,","); a=a+1; if(m[1]==n, while(a%3>0,a++); m=if(length(m)==1,[],vecextract(m,"2..")),if(a%3==0,a++)); m=concat(m,a))}

a(a(n)) = 3*(n+4).

More terms and PARI code from Klaus Brockhaus, Mar 07 2003

A079258 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a square".

Original entry on oeis.org

0, 1, 3, 4, 9, 10, 11, 12, 13, 16, 25, 36, 49, 64, 65, 66, 81, 82, 83, 84, 85, 86, 87, 88, 89, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153

Offset: 0

N. J. A. Sloane and Matthew Vandermast, Feb 04 2003

Also, a(n) is smallest nonnegative integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = n^2.

Ivan Neretin, Table of n, a(n) for n = 0..10000
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)
Index entries for sequences of the a(a(n)) = 2n family

See A079000, A079253, A079254, A079256, A079257 for similar sequences.

a = {1, 3}; Do[AppendTo[a, If[MemberQ[a, n], Position[a, n][[1, 1]]^2, a[[-1]] + 1]], {n, 3, 58}]; Prepend[a, 0] (* Ivan Neretin, Jul 09 2015 *)

A080639 a(1) = 1; for n>1, a(n) is taken to be the smallest integer greater than a(n-1) which is consistent with the condition "for n>1, n is a member of the sequence if and only if a(n) is even".

Original entry on oeis.org

1, 2, 5, 7, 8, 9, 10, 12, 14, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 30, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 96

Offset: 1

N. J. A. Sloane and Benoit Cloitre, Feb 28 2003

Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)
Index entries for sequences of the a(a(n)) = 2n family

Cf. A079253, A079000.

{a(a(n))} = {1, 2, 2i, i >= 4}.

More terms from Matthew Vandermast, Feb 28 2003

A079256 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a power of 2".

Original entry on oeis.org

1, 2, 5, 6, 8, 16, 17, 32, 33, 34, 35, 36, 37, 38, 39, 64, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 32769, 32770, 32771, 32772, 32773, 32774, 32775, 32776, 32777, 32778

Offset: 1

N. J. A. Sloane and Matthew Vandermast, Feb 04 2003

Robert Israel, Table of n, a(n) for n = 1..10000
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

See A079000, A079253, A079254 for similar sequences.

A:= Vector(100):
A[1..5]:= <1,2,5,6,8>:
going:= true;
for n from 3 while going do
  for k from 0 to A[n+1]-A[n] do
    if A[n]+k > 100 then going:= false; break fi;
    A[A[n]+k]:=  2^n+k
od od:
convert(A,list); # Robert Israel, Mar 20 2019

A080710 a(0) = 1; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 3".

Original entry on oeis.org

1, 3, 4, 6, 9, 10, 12, 13, 14, 15, 18, 19, 21, 24, 27, 30, 31, 32, 33, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 54, 57, 58, 59, 60, 63, 64, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 105, 108, 111, 112, 113, 114, 117, 118, 120

Offset: 0

N. J. A. Sloane, Mar 05 2003

Is this the same sequence as A115837? - Andrew S. Plewe, May 08 2007

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)
Index entries for sequences of the a(a(n)) = 2n family

Cf. A003605, A079253, A080711, A080712.

Cf. A079000, A003605, A079253, A080711, A080712.

{a=1; m=[1]; for(n=1,67,print1(a,","); a=a+1; if(m[1]==n, while(a%3>0,a++); m=if(length(m)==1,[],vecextract(m,"2.."))); m=concat(m,a))}

a(a(n)) = 3*(n+1).

More terms and PARI code from Klaus Brockhaus, Mar 06 2003

A379051 Lexicographically earliest infinite sequence of distinct positive numbers with the property that n is a member of the sequence iff a(n) is composite.

Original entry on oeis.org

2, 4, 5, 6, 8, 9, 10, 12, 14, 15, 7, 16, 17, 18, 20, 21, 22, 24, 23, 25, 26, 27, 28, 30, 32, 33, 34, 35, 31, 36, 38, 39, 40, 42, 44, 45, 41, 46, 48, 49, 50, 51, 47, 52, 54, 55, 56, 57, 58, 60, 62, 63, 59, 64, 65, 66, 68, 69, 70, 72, 67, 74, 75, 76, 77, 78, 80

Offset: 1

N. J. A. Sloane, Dec 17 2024

nonn

The sequence tells you exactly which terms of the sequence are composite: the second, fourth, fifth, sixth, etc. terms are composite, and this is the lexicographically earliest sequence with this property.
Let P be a property of the nonnegative integers, such as being a prime.
The OEIS contains many entries whose definitions have the following form.
"The sequence is the lexicographically earliest infinite sequence of distinct positive (or sometimes nonnegative) integers with the property that n is a term of the sequence iff a(n) has property P."
That is, the terms of the sequence tell you which terms of the sequence have the property. A121053 is the classical example.
Since these are lists, the offset is usually 1.
There are two versions, one where the sequence is required to be strictly increasing, and an unrestricted version which is not required to be increasing.
Examples:
  Property P    Unrestricted    Increasing
  ----------------------------------------
  Prime         A121053         A079254, A334067 (offset 0)
  Even          A080032         A079253
  Odd           A079313         A079000
  Composite     A379051         A099797
  Not composite A377901         A099798
  Not prime     A379053         A085925

Michael De Vlieger, Table of n, a(n) for n = 1..65536 [Terms 1 to 10000 from Scott R. Shannon]

Cf. A079000, A079253, A079254, A079313, A080032, A085925, A099797, A099798, A121053, A334067, A377901, A379053.

A379053 Lexicographically earliest infinite sequence of distinct positive numbers with the property that n is a member of the sequence iff a(n) is not a prime.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 9, 10, 12, 14, 13, 15, 16, 18, 20, 21, 19, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 23, 36, 37, 38, 39, 40, 42, 44, 45, 46, 48, 49, 43, 50, 51, 52, 54, 55, 53, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 61, 70, 72, 74, 75, 76, 77, 78, 71

Offset: 1

N. J. A. Sloane, Dec 17 2024

nonn

The sequence tells you exactly which terms of the sequence are either 1 or composite.

See the Comments in A379051 for further information.

Michael De Vlieger, Table of n, a(n) for n = 1..65536 [Terms 1 to 10000 from Scott R. Shannon]

Cf. A079000, A079253, A079254, A079313, A080032, A085925, A099797, A099798, A121053, A334067, A377901, A379051.

nn = 120; u = 3; v = {}; w = {}; c = 4;
{1}~Join~Reap[Do[
  If[MemberQ[w, n],
    k = c; w = DeleteCases[w, n],
    m = Min[{c, u, v}]; If[And[CompositeQ[m], n < m],
      AppendTo[v, n]];
      If[Length[v] > 0,
        If[v[[1]] == m,
        v = Rest[v]]]; k = m];
    AppendTo[w, k]; If[k == c, c++; While[PrimeQ[c], c++]]; Sow[k];
If[n + 1 >= u, u++; While[CompositeQ[u], u++]], {n, 2, nn}] ][[-1, 1]] (* Michael De Vlieger, Dec 17 2024 *)

When sorted, this appears to be the complement of [2, 5, 11, 17, 29, and prime(2*t+1), t >= 35]. - Scott R. Shannon, Dec 18 2024

More terms from Michael De Vlieger, Dec 17 2024

A079257 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a triangular number".

Original entry on oeis.org

0, 1, 4, 5, 6, 10, 15, 16, 17, 18, 21, 22, 23, 24, 25, 28, 36, 45, 55, 56, 57, 66, 78, 91, 105, 120, 121, 122, 136, 137, 138, 139, 140, 141, 142, 143, 153, 154, 155, 156, 157, 158, 159, 160, 161, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 190, 210, 231

Offset: 0

N. J. A. Sloane and Matthew Vandermast, Feb 04 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

See A079000, A079253, A079254, A079256, A079258 for similar sequences.

A079259 a(n) is taken to be the smallest positive integer greater than a(n-1) such that the condition "n is in the sequence if and only if a(n) and a(n+1) are both odd" can be satisfied.

Original entry on oeis.org

1, 5, 6, 10, 11, 15, 19, 20, 24, 25, 29, 33, 34, 38, 39, 43, 44, 48, 49, 53, 57, 58, 62, 63, 67, 71, 72, 76, 77, 81, 82, 86, 87, 91, 95, 96, 100, 101, 105, 109, 110, 114, 115, 119, 123, 124, 128, 129, 133, 137, 138, 142, 143, 147, 148, 152, 153, 157, 161, 162, 166, 167, 171, 175, 176, 180, 181, 185, 186, 190, 191, 195, 199, 200, 204

Offset: 1

N. J. A. Sloane and Matthew Vandermast, Feb 04 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A079000, A079253-A079258.

A080711 a(0) = 2; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 3".

Original entry on oeis.org

2, 4, 6, 7, 9, 10, 12, 15, 16, 18, 21, 22, 24, 25, 26, 27, 30, 31, 33, 34, 35, 36, 39, 40, 42, 45, 48, 51, 52, 53, 54, 57, 58, 60, 63, 66, 69, 70, 71, 72, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 90, 93, 96, 97, 98, 99, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112

Offset: 0

N. J. A. Sloane, Mar 05 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)
Index entries for sequences of the a(a(n)) = 2n family

Cf. A079000, A003605, A079253, A080710, A080712.

{a=2; m=[2]; for(n=1,68,print1(a,","); a=a+1; if(m[1]==n, while(a%3>0,a++); m=if(length(m)==1,[],vecextract(m,"2..")),if(a%3==0,a++)); m=concat(m,a))}

a(a(n)) = 3*(n+2).

More terms and PARI code from Klaus Brockhaus, Mar 06 2003

cp's OEIS Frontend

A080720 a(0) = 5; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 3".

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A079258 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a square".

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A080639 a(1) = 1; for n>1, a(n) is taken to be the smallest integer greater than a(n-1) which is consistent with the condition "for n>1, n is a member of the sequence if and only if a(n) is even".

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A079256 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a power of 2".

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Maple

A080710 a(0) = 1; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 3".

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Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A379051 Lexicographically earliest infinite sequence of distinct positive numbers with the property that n is a member of the sequence iff a(n) is composite.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A379053 Lexicographically earliest infinite sequence of distinct positive numbers with the property that n is a member of the sequence iff a(n) is not a prime.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A079257 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a triangular number".

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A079259 a(n) is taken to be the smallest positive integer greater than a(n-1) such that the condition "n is in the sequence if and only if a(n) and a(n+1) are both odd" can be satisfied.

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